(since kk is assumed to be of characteristic 0). Under this equivalence we have that U∈cAlgk↪cAlgkΔop↪HU \in cAlg_k \hookrightarrow cAlg_k^{\Delta^{op}} \hookrightarrow \mathbf{H} is 𝒪\mathcal{O}-perfect:

is simply the formal dual of UKU^K, which is K⋅𝒪(U)K \cdot \mathcal{O}(U) formed in cdgAlgkcdgAlg_k by formal duality. By the above proposition the inclusion cdgAlgk−↪cdgAlgkcdgAlg_k^- \hookrightarrow cdgAlg_k preserves this (∞,1)(\infty,1)-colimit.